Three Dimensions in Motivic Quantum Gravity

Three Dimensions in Motivic Quantum Gravity M. D. Sheppeard December 2018 Abstract Important polytope sequences, like the associahedra and permutohedra, contain one object in each dimension. The more particles labelling the leaves of a tree, the higher the dimension required to compute physical amplitudes, where we speak of an abstract categorical dimension. Yet for most purposes, we care only about low dimensional arrows, particularly associators and braiding arrows. In the emerging theory of motivic quantum gravity, the structure of three dimensions explains why we perceive three dimensions. The Leech lattice is a simple consequence of quantum mechanics. Higher dimensional data, like the e8 lattice, is encoded in three dimensions. Here we give a very elementary overview of key data from an axiomatic perspective, focusing on the permutoassociahedra of Kapranov. These notes originated in a series of lectures I gave at Quantum Gravity Re- search in February 2018. There I met Michael Rios, whom I originally taught about the associahedra and other operads many, many years ago. He discussed the magic star and exceptional periodicity. In March I met Tony Smith, the founder of e8 approaches to quantum gravity. In June I attended the Advances in Quantum Gravity workshop at UCLA, where I met the M theorist Alessio Marrani, and in July participated in Group32 in Prague, where I met Piero Truini, who I later talked to about operad polytopes in motivic quantum gravity. All these activities led to a greatly improved understanding of the emerging theory. I was then forced to return to a state of severe poverty, isolation and abuse, ostracised effectively from the professional community, from local com- munities, from family, and many online websites. I am working now on an ipad in my tent. Although I hope the discussion here is elementary, the motivation for it comes from deep mathematical conundrums, as any good question in quantum gravity should. As a category theorist, the slow development of topological field theories was endlessly frustrating, with its separation of spatial and algebraic data. Like in gauge theories, one was supposed to mark spatial diagrams with objects from an algebraic category, such as a traditional category of Hilbert spaces. But the most interesting functors are monadic, and it seemed that gravity should canonically interpret its geometry in an algebraic way. Moreover, the common habit of grabbing R or C as a God given set of numbers was in 1 conflict with both the axioms of a topos [1] and the need to construct classical manifolds from discrete quantum data. We begin with the power set monad [1], which simply maps any set to the set of all subsets of the set. In the case of a three element set, there are eight possible subsets including the empty set, listed by the vertices of figure 1. A set with n elements gives a cube in dimension n. An edge on a cube, directed away from the empty set 1, is an inclusion of sets. The concatenation of elements, like αiαj, denotes the union. Conversely, reverse arrows restrict to a subset of a set, and the source of a square is the intersection of sets. A category theorist talks about the duality of reversing arrows, or the opposite category of the category of all sets. But quantum mechanics is not about sets. Axioms based on set theoretic mathematics must be replaced by axioms for quantum logic. In the next section we see that n-cubes also play an important role in the quantum case. The signed vertices on a cube denote a state of n qubits in the theory of quantum computation [2][3][4], and we use the universality of special categories like the Fibonacci anyons [5][6] to argue that gauge groups emerge from braid group representations. Figure 1: The parity cube in dimension 3 From cubes we move on to the associahedra [7] and permutohedra [8], the polytopes that are specified by rooted binary trees or permutations. The combi- nation of these polytopes, including both Mac Lane pentagons [9] and braiding hexagons, are the permutoassociahedra of [10]. In particular, the 120 vertex permutoassociahedron in dimension 3 stands in for half the 240 roots of the 8 dimensional e8 lattice. So what is a motive? Certainly, it does not begin with a variety based on C, or any other random field. A whole, infinite dimensional category of motives 2 should be linked to our quantum axioms, taking motivic diagrams directly to physical results, following the original vision of Grothendieck [11]. A classical spacetime is a mere afterthought, a three time Minkowski space perhaps com- pactified to SU(2) × SU(2), constructed using the anyon braid group B3 [12]. Ribbon diagrams are spatial, and integral octonions, for instance, are algebraic, but the category of motives thinks they are one and the same, thinks that a fun- damental group drawn in knots is its own algebraic structure. And the Standard Model of particle physics lives here. 1 A motivic perspective Geometric categorical axioms encode algebraic information. In the theory of quantum gravity, we require a canonical symbiosis of discrete geometry and algebraic rules. Motivated by the power set monad [1], we begin with the humble cube. The vertices of figure 1 carry two sets of labels. Physically, the signs encode the fractional charges of the anyons [13] underlying the leptons and quarks, so that −−− denotes 3 zero charges on a neutrino and +++ denotes the charge on a positron. There is a second cube for the negative leptons and quarks [14][15]. In quantum computation, three signs stand for a set of 3 qubit states. At the same time, the edges of a 3-cube move in the 3 directions of a qutrit space. A qutrit has three possible measurement outcomes, while a qubit has two. A path from the source 1 to the target α1α2α3 has three edges. These six paths clearly form a copy of the permutation group S3, if we label the directions 1, 2 and 3. Seven of the vertices give an unconventional Fano plane, as shown in figure 2. This gives us a basis for the integral octonions [16][17], which are appearing here automatically, once we decide to study sets carefully. In quantum mechanics, an element of a set usually becomes a basis vector in a state space. Instead of three points and their unions, consider three generic lines in a Euclidean plane with respect to intersection in the plane. If no lines are selected, we have the whole plane. If one line is selected, we get a line. If two lines are selected, we obtain the point of intersection of the lines. And finally, if all three lines are selected, we have the empty set. Observe how these possibilities reverse the position of empty set, moving it from 1 to the target of the cube. The source 1 is now the entire plane, an axiomatic object of dimension 2. The singleton αi are now dimension 1 objects and the αiαj of dimension zero. Whereas sets are always zero dimensional, whatever the cardinality, quantum geometry interprets a whole number n as a dimension. In this example, there are spaces of dimension 0, 1 and 2. A classical topolo- gist would take these numbers and put them on the vertices of a triangle, calling the triangle a 2-simplex. We have already drawn the 2-simplex as the interior of three lines in a plane. It is also the diagonal slice of the 3-cube at the αi posi- tions, or rather, the points 001, 010 and 100. Now consider qutrit words which allow more than one step in each direction. As well as 231 2 S3 we have 2231, taking two steps in one direction. Allowing arbitrary noncommutative words, 3 Figure 2: The Fano plane we see discrete paths on rectangular blocks in any dimension. On a cubic array of paths, the diagonal slice fixes the length of the word, so that length 3 qutrit words collapse to the discrete simplex of figure 3. Simplices carry commutative words, while cubes carry the full set of noncommutative words. Given any noncommutative word, a rectangular subspace of subwords sits below it. For example, 2231 sits on a 3-cube with three points along each edge, while its subpath 231 lies in the space of paths closer to the source vertex. 2 If the stepping represents powers of a set of primes, as in p2p3p1, then the subspace vertices list the divisors, and each dimension out to 1 lists powers of an integral prime. Then the positive integers in Z use up every cubical point in every dimension. A qubit corresponds to the prime 2 and a qutrit to the prime 3, because state spaces are characterised by Schwinger's mutually unbiased bases [18][19][20]. For qubits, the three Pauli matrices are replaced by the three unitary bases of eigenvectors, 1 1 1 1 1 i 1 0 F2 = p ;R2 = p ;I2 = : (1) 2 1 −1 2 i 1 0 1 4 Figure 3: The commutative tetractys for three qutrits For qutrits, the bases are 01 1 11 01 ! 11 01 1 !1 1 1 −1 1 F3 = p @1 ! !A ;R3 = p @1 1 !A ;R3 = p @! 1 1A ;I3; 3 1 !! 3 ! 1 1 3 1 ! 1 (2) where ! is the cubed root of unity. In every prime power dimension d, there is a (one step) circulant generator Rd whose powers give d bases that are mutually unbiased. Thus each cubic point is associated to a set of d = pr matrices. Dimension 3 is special because circulants are Hermitian, like the elements of the exceptional Jordan algebra over the octonions [16][17].

Load more